3.01 Pythagoras' theorem

Pythagoras' theorem tells us that the three sides of any right-angled triangle (a triangle that has a $90^\circ$90° angle) are related in a simple way.

Let's take a general right-angled triangle with sides measuring $a$a, $b$b and $c$c:

The sides $a$a, $b$b and $c$c are related in the following way:

This tells us that the square of the hypotenuse is equal to the sum of the squares of the two shorter sides:

So, if any two sides of a triangle are known and one side is unknown, this relationship can be used to find the length of the unknown side. 

We can always rearrange the equation any which way to make the unknown side the subject:

$c=\sqrt{a^2+b^2}$c=√a2+b2

$b=\sqrt{c^2-a^2}$b=√c2−a2

$a=\sqrt{c^2-b^2}$a=√c2−b2

Is the triangle right-angled?

The theorem can also be used to determine whether a triangle is right-angled.

For example, if a triangle has side lengths measuring $6$6 cm, $8$8 cm and $10$10 cm, we can check if it satisfies Pythagoras' theorem:

$c^2=a^2+b^2$c2=a2+b2

$RHS$RHS$=$=$10^2=100$102=100   and  $LHS$LHS$=$=$6^2+8^2=100$62+82=100

So $10^2=6^2+8^2$102=62+82, and a triangle with sides $6$6, $8$8 and $10$10 units is a right-angled triangle.

 

Worked examples

Question 1

A right-angled triangle has two shorter sides $10$10 cm and $12$12 cm. Find the length of the hypotenuse.

In Pythagoras's theorem, $c^2=a^2+b^2$c2=a2+b2, we want to find $c$c, and are given $a$a and $b$b.

Let's substitute the values in for $a$a and $b$b and then solve for $c$c.

$c^2$c2	$=$=	$10^2+12^2$102+122
$c$c	$=$=	$\sqrt{10^2+12^2}$√102+122
$c$c	$=$=	$15.62$15.62 (rounded to two decimal places)

So, the hypotenuse is approximately $15.62$15.62 cm long.

 

Question 2

 

Question 3

 

Question 4

 

Question 5